Read this response and predict the preceding dialog. The associated matrices and their inverses for this definition of the index sets formula_105 then allow us to perform the analog of Moebius inversion for divisor sums which can be used to express the summand functions "f" as a quasi-convolved sum over the inverse matrix entries and the left-hand-side special functions, such as formula_106 or formula_20 pointed out in the last pair of examples.

Phone call:
P1: What are the different applications of the Redheffer matrix?.
P2: Besides non-trivial estimates on the corresponding spectrum and eigenspaces associated with truly notable and important applications of these representations -- the underlying machinery in representing sums of these forms by matrix products is to effectively define a so-termed masking matrix whose zero-or-one valued entries denote inclusion in an increasing sequence of sets of the natural numbers formula_97..
P1: What other applications are there?.
P2: To illustrate that the previous mouthful of jargon makes good sense in setting up a matrix based system for representing a wide range of special summations, consider the following construction: Let formula_98 be a sequence of index sets, and for any fixed arithmetic function formula_99 define the sums One of the classes of sums considered by Mousavi and Schmidt (2017) defines the relatively prime divisor sums by setting the index sets in the last definition to be This class of sums can be used to express important special arithmetic functions of number theoretic interest, including Euler's phi function (where classically we define formula_102) as   and even the Mobius function through its representation as a discrete (finite) Fourier transform: Citations in the full paper provide other examples of this class of sums including applications to cyclotomic polynomials (and their logarithms)..
P1: Is there anything else interesting?.
P2: The referenced article by Mousavi and Schmidt (2017) develops a factorization-theorem-like treatment to expanding these sums which is an analog to the Lambert series factorization results given in the previous section above..
P1: What other uses are there for the Redheffer matrix?.
P2: 